Computing accurate splines of degree greater than three is still a challenging task in today's applications. In this type of interpolation, high-order derivatives are needed on the given mesh. As these derivatives are rarely known and are often not easy to approximate accurately, high-degree splines are difficult to obtain using standard approaches. In Beaudoin (1998), Beaudoin and Beauchemin (2003), and Pepin et al. (2019), a new method to compute spline approximations of low or high degree from equidistant interpolation nodes based on the discrete Fourier transform is analyzed. The accuracy of this method greatly depends on the accuracy of the boundary conditions. An algorithm for the computation of the boundary conditions can be found in Beaudoin (1998), and Beaudoin and Beauchemin (2003). However, this algorithm lacks robustness since the approximation of the boundary conditions is strongly dependant on the choice of $\theta$ arbitrary parameters, $\theta$ being the degree of the spline. The goal of this paper is therefore to propose two new robust algorithms, independent of arbitrary parameters, for the computation of the boundary conditions in order to obtain accurate splines of any degree. Numerical results will be presented to show the efficiency of these new approaches.
翻译:计算大于三阶的精确样条仍然是当今应用中具有挑战性的任务。在这种插值中,需要使用给定网格上的高阶导数。由于这些导数很少被了解,并且往往不容易准确地逼近,因此使用标准方法很难获得高次样条。在 Beaudoin(1998)、Beaudoin和Beauchemin(2003)和Pepin等人(2019)中,分析了一种从等距插值节点基于离散傅里叶变换计算低次或高次样条近似值的新方法。该方法的精度在很大程度上取决于边界条件的精度。可以在Beaudoin(1998)和Beaudoin和Beauchemin(2003)中找到计算边界条件的算法。然而,由于边界条件的逼近强烈依赖于$\theta$这个任意参数的选择,$\theta$是样条的阶数,因此该算法缺乏稳健性。因此,本文旨在提出两种新的稳健算法,独立于任意参数,用于计算边界条件,以获得任意阶数的精确样条。将呈现数值结果以展示这些新方法的效率。